Abstract

We establish the property of completeness in the space \(L_2(D) \), where \(D \) is a domain in \(\mathbb {R}^n \), \(n\geq 3\), of the products of all possible regular solutions of the equation \(\Delta u-\kappa ^2 u=0 \), \(\kappa \in \mathbb {R}_{+}\cup i\mathbb {R}_{-} \), by the fundamental solutions of this equation with singularities on a straight line that does not meet \(\overline {D}\). This result is used when establishing the uniqueness of the solution of a coefficient inverse problem of wave tomography in a nonoverdetermined statement.

中文翻译：

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二阶椭圆方程解的不对称乘积的完备性与波动方程反问题的解的唯一性

摘要

我们在空间\（L_2（D）\）中建立完整性的性质，其中\（D \）是\（\ mathbb {R} ^ n \），\（n \ geq 3 \）的域方程\（\ Delta u- \ kappa ^ 2 u = 0 \），\（\ kappa \ in \ mathbb {R} _ {+} \ cup i \ mathbb {R} _ {-} \），通过不等式在不满足\（\ overline {D} \）的直线上具有奇点的方程的基本解。在未确定的陈述中建立波动层析成像系数反问题的解的唯一性时，将使用此结果。